A Generic Approach to Iterative Methods

Abstract

Iterative processes usually appear in the undergraduate curriculum as unrelated topics, each having a specific purpose and independent existence. Yet such topics as Newton's method, Picard's method for showing the existence and uniqueness of solutions to initial value problems, the Economic Cobweb Theorem for describing supply and demand equilibria, the convergence theorem for the long-run behavior of regular Markov chains, as well as many other iterative processes, are simply different manifestations of the general theme set forth in the Contraction Mapping Principle. The first abstract setting for this principle is credited to Stefan Banach [2],[17],[27], who showed that, under a very general hypothesis, all sequences generated by the repeated evaluation of a distance-decreasing function must converge to a unique fixed point. This convergence is the essence of an iterative technique which can be used in a variety of applications to find an approximate solution, to assert that a unique solution must exist, or to show that a given sequence converges to a known solution. Such applications of the Contraction Mapping Principle are the substance of this article. A good example of a simple iterative process was shown to me by my five-year-old son Donald: Turn on a scientific calculator and repeatedly press the COS button. On a typical calculator, this algorithm computes iterated values of the calculator function COS(x), degree mode (a discrete rational function that approximates the continuous real function f(x)= cos(2 gx/360)), and displays successive terms of the sequence